Differential Invariants of the Kerr Vacuum

TL;DR

This paper investigates the behavior of gradient norms of invariants in the Kerr vacuum, revealing how they relate to the black hole's angular momentum and extending findings to Kerr-Newman geometry.

Contribution

It provides a detailed analysis of the global behavior of gradient invariants in Kerr and Kerr-Newman spacetimes, highlighting their relation to angular momentum and horizon properties.

Findings

Gradient norms reflect angular momentum by decreasing with increasing spin.

Both invariants identify horizons locally but differ globally.

Results extend to Kerr-Newman geometry, linking invariants to charge and rotation.

Abstract

The norms associated with the gradients of the two non-differential invariants of the Kerr vacuum are examined. Whereas both locally single out the horizons, their global behavior is more interesting. Both reflect the background angular momentum as the volume of space allowing a timelike gradient decreases with increasing angular momentum becoming zero in the degenerate and naked cases. These results extend directly to the Kerr-Newman geometry.